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geometry 4d

Hopf fibration fibres as linked circles

Lift base points of S2 to their Hopf fibres in S3, project them stereographically into linked circles, check the fibration's properties (great circles, Clifford parallels, linking number 1, tori over latitude circles), and write tubes plus a colour-matched base sphere as glTF.

Recorded render of Hopf fibration fibres as linked circles

What this component contains

  • ATTRIBUTION.md
  • LICENSE
  • README.md
  • component.json
  • example.gltf
  • fourd.py
  • hopf_fibration.py
  • preview.png
  • test_package.py
  • verification/native.json

How it works

Entry points
hopf_map, lift, fibre, base_points, colour, is_great_circle, is_circle_3d, linking_number and 6 more
Outputs
example.gltf
Coordinates
quaternion a + bi + cj + dk at (x, y, z, w) = (b, c, d, a); stereographic image from (0, 0, 0, 1), Godot/glTF +Y up

Recorded checks

godot 4.7.2.stable.official.ed1daf0bf

  • cli_rebuilds_example
  • gltf_structure_example
  • cli_example_runs
  • cli_example_gltf_structure
  • godot_loads_example
  • godot_loads_cli_clifford_fibres
  • preview_render

Limits to consider

Fibres are polygons of the given sample count, so their images are polygonal circles. Base circles must avoid the x axis direction, whose fibre passes through the projection pole and becomes a line; fibres near it become very large. The inset base sphere uses alpha blending.